These high-level conditions will be used to prove convergence of \(\theta _{b}\)  in \(L_p\) -norm, \(p \ge 2\) . Appendix  provides sufficient conditions for i-ii. using either \(m\)  out of \(n\)  resampling with replacement or reweighting with Gaussian multiplier weights.Dependence of \(C_2,C_3\)  on \(n\)  is omitted to simplify notation. The conditions under Gaussian reweighting are the same for \(p=2\)  and \(p>2\)  but not with resampling. The results can be extended to other weighting distributions with the same conditions as resampling by adapting the proof of Theorem 3.6.13 in [1]} similar to [2]}. Consistent estimation and valid inference using quantiles in Algorithm  (steps 4 and 5), will require \(p=2\)  in Assumption REF . Consistency of standard errors further requires \(p=4\) . Consistency of moments for the standard bootstrap is also associated with particular moment conditions, see [3]}, [4]}, [2]}. Since the estimator in Algorithm  is computed by averaging draws, the derivations are fundamentally moment-based.
